Hit Danish TV series Forbrydelsen (The Killing) might be done and dusted, but the fashion legacy of the famous Faroese jumper lives on. Not since Dorothy was paired with the red shoes has protagonist and costume been so well matched as Sarah Lund and that jumper.

It was back in 2007 that actor Sofie Grabol and a costume designer from the Danish Broadcasting Corporation came across a modern take on the traditional Faroese fisherman's star-motif jumper by the label Gudrun & Gudrun and decided it would be perfect for the wardrobe of phlegmatic detective Lund. The character wore variations on the label's signature knitwear in all three of the show's series, and the jumper became as talked about as the show's plotline. The knock-on effect was that this tiny company, based in the unlikely fashion territory of the Faroe Islands in the North Atlantic, became synonymous with an international award-winning TV hit.

The label, by the way, has no wish to be defined by a single piece of knitwear. To the two founders, Gudruns Rogvadottir and Ludvig, it was always just a great jumper. Gudrun & Gudrun are champions of what's known as eco luxe, defined by Vogue Italia editor Franca Sozzani, among others, as the new luxury in which authenticity, craftsmanship and elegance create value. Its clothes are mainly handmade by Faroese and Jordanian women (traditional Jordanian knitwear is created in similar ways to those in the Faroe Islands) and its classic jumper range uses 100% untreated and undyed wool.

On the case: (from left) Sofie Grabol; Camilla, Duchess of Cornwall, and Crown Princess Mary on the set of The Killing. Photograph: Rex Features/Tim Rooke

The brand is worlds away from the high-street fast fashion that is so distanced from (and often indifferent to) its supply chain. And it takes a slow-fashion approach: it has dropped the idea of a summer collection because their hearts weren't really into lightweight cotton knitwear for the sake of making more product. In an industry where every other brand is furiously adding extra collections – resort, preseason – to flog increasing mounds of product, this mindset is unusual.

Nor are there many brands that carry out their own fashion harvest. I found myself on a boat heading to the island of Koltur (number of inhabitants: two, and one is part-time) to witness this first hand. There are approximately 70,000 Faroese sheep (descended, we think, from wild sheep) well outnumbering the population of 45,000 people living across the 18 islands. Evidence suggests that both species have coexisted here since at least the 8th century.

The usual annual window for shearing is early July, but this summer the Gulf Stream wasn't playing ball, the mists descended and Koltur became inaccessible. So this is the first day the gaggle of family and friends has been able to get out to hand-shear the sheep, which must first be chased down the mountain (I'm beginning to realise there are no easy jobs on the Faroe Islands, which are both relentlessly beautiful and tough). The wool will be sent to Scotland for washing and to Lithuania for spinning. Gudrun & Gudrun will use about a tonne in its collection.

The wool was once known as Faroese gold, but by the 1990s, in common with European communities, the price for fleece had tumbled so far that families were forced to burn it rather than paying to wash and process it.

Rogvadottir was raised in the Faroes but had moved to Denmark, where she worked in international development. When she heard about the burning while on a visit to see her parents, she was outraged. So, back in 1999, she asked her friend Gudrun Ludvig, who she knew to be a skilled designer, to use the wool to make a few pieces.

At that point Ludvig, who had trained as a designer in Denmark but was born and raised on Suðuroy, the southernmost island, found herself a single mother in need of employment and a creative outlet. The company became a major force when the two gave up their jobs to concentrate on the label in 2007.

Once the brand got going, the ingredient it needed, next to raw fibre, was excellent knitters. Again, the Faroes delivered. Around 40 women (many relatives and family friends) knit for Gudrun & Gudrun across the islands. They knit around their daily lives, in between other jobs and running the home or animals, in an arrangement most similar to crofting. They are sustained by the extra money and the outlet for their skill, and are enormously proud of contributing to the brand.

An especially trusted clique of knitters decipher Ludvig's designs and help her convert them into replicable patterns. They are technically skilled problem solvers. They are often experienced in knitting the precise, intricate, knitted blouses that are an important part of Faroese national dress.

Island life: the landscape of the Faroes is a world away from fast fashion. Photograph: Jamie Baker for the Observer Jamie Baker/Observer

Traditional garments are an inspiration for the Gudrun & Gudrun aesthetic, but the designs are definitely fashion led. The top floor of the company's cute wooden Torshavn store is given over to Ludvig's design studio, where she's experimenting with techniques and stitches and a couture piece for a Paris show. The brand is as rooted in history and craft production as any Paris fashion house, but at Gudrun & Gudrun the emphasis is always about moving the aesthetic on – and their ambitions extend far beyond the Faroes.

The label's fanbase stretches around the world and its designs are stocked as far afield as Japan and Korea. Perhaps the only country in which it has yet to develop a real foothold is the US (notably Sarah Linden, the detective in the American remake of The Killing, wore a succession of Fairisle- lite jumpers, only succumbing to a real Gudrun & Gudrun in the second series).

"American styling is very different," says Rogvadottir. "It seems to be a lot about wearing black dresses with thin black belts and looking as small as possible."

I end my visit with a mandatory dip in the North Atlantic at the outdoor swimming pool where Ludvig swims most days. If ever there was a brand that embodies the Viking-adventurer spirit, this is it. It makes few concessions to the faint-hearted.

I note one – an alpaca version of the Lund jumper created for consumers who love the idea but can't stomach the itch of the original Faroese wool or find it a little bulky (perhaps they're American). Personally, I prefer the original. It's unbelievably warm, and the natural fibre is water resistant (precisely why generations of Faroese fishermen wore similar handknits). It also smells unmistakably sheepy (that's the lanolin). Besides, even though I know Gudrun & Gudrun offers so much more than a moment of TV fame, it still makes me think of Lund.